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Review

So what is a function?

A function is a relation that for every input, there is only one output.

What is a relation?

A relation is a set of ordered pairs

So a function is a set of ordered pairs that has only one output for each input.

Of course this is a review from Chapter 4, when we studied functions, however there are a few more things about functions that we just touched on, and now we’ll go into more detail.

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WHAT A FUNCTION REALLY IS

Think of a function like a machine.

Let’s say you have a machine that makes, say, smart phones, and all it needs to create the smart phones is some matter of some sort (let’s say, tumbleweeds).

So you take a bunch of tumble weeds, plug them into the machine, and get a smart phone out. Seems simple enough right?

So you go out and gather as many tumbleweeds as you can (because it’s California and they’re freaking everywhere) and you start shoving them into the machine.

But, you realize that the more raw material you shove into this machine, the more you get out, and since this is a machine that only makes smart phones, you know that if you put in 70 tumbleweeds, you’re not going to get back a tablet or laptop, you’re going to get back 1 smart phone.

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Essentially, this is a function

That’s basically what a function is.

A function is a machine that spits out only one number for each number that is put in.

So, using our machine analogy, imagine if you were trying to use the phone making machine one day. You push 100 tumbleweeds into the machine (very carefully, because ow), you push start, hear the usual whirring of the machine, and think you’re going to get out a few smart phones.

However, this time around, instead of getting a smart phone, you get a vhs tape.

(For those of you who don’t know what that is, it’s like a DVD but older.)

Now, are you going to put your faith in that machine from here on out?

What if, from now on you get nothing but VHS tapes, or even worse, it starts spitting out even older things instead? (Like an Atari, 8 bit player, record player, or even worse.

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SAME CONCEPT HERE

If your machine that is only suppose to spit out smart phones, spits out something else instead, it’s no longer a smart phone machine right?

Same thing, if you run into something that seems like a function but putting in a single number can give you two answers, it’s no longer a function.

So now the question becomes, why are there so many different functions out there?

Let’s revisit our analogy.

I purposefully only used a generic “smart phone” because there are tons of different smart phones out there right?

So what if your machine made old Samsung Galaxy S3’s (yes, they do exist) but you wanted it to make the Iphone Xr?

Then you need to change up the inner mechanisms of your machine right?

Same idea, there are tons of different functions because there are tons of different problems that need to be solved.

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SO WHY DO YOU NEED TO KNOW THIS?

Mainly because we use functions constantly.

When you’re in the hospital, there’s a function that controls the monitor that watches your heart beat.

At the power plant, there’s a function that monitors how much electricity goes into your home (so it doesn’t over load and catch on fire).

On you’re phone, there are algorithms which are nothing more than functions within functions (functcetpion!)

There are functions that tell the government how much they owe you back when you do your taxes (the better the function, the more money in your pocket!)

There are functions in your car that tell you how fast you’re going so the cop that’s a mile down the road can’t ticket you.

There are functions in the speed gun the cop is using to pull you over.

There are functions everywhere, in almost every part of your life.

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SO HOW DO WE SOLVE FUNCTIONS?

First thing we need is the function. We can’t solve something that isn’t there.

Usually, a function is some sort of letter, with parenthesis next to it and a letter in the parenthesis(What we call function notation).

So something like: f(x) = 3x + 5

Now what we do is plug things in for x (add the tumbleweeds).

So f(3) would be something like:

f(3) = 3(3) + 5

f(3) = 9 + 5

f(3) = 14

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What if we know what f(x) is?

There will be times when we know what the answer to the function is, but we need to know what made that answer (kind of like how a phone repair guy will put an app on your phone, knowing what it should do to fix it.)

So in our example: f(x) = 3x + 5, let’s say they give you f(x) = 5, then what we do is:

Substitute!

So:

f(x) = 3x + 5 becomes

5 = 3x + 5

We subtract 5 and get

0 = 3x

Divide by 3 and x = 0.

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SO WHY USE F(X) INSTEAD OF JUST Y?

Now let’s say they want you to find out what g(2) is. �All we do is look at where the graph lies when x = 2.

And now we see that when x = 2, g(x) = 4.

So g(2) = 4

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THERE ARE SOME RULES WHEN IT COMES TO FUNCTIONS HOWEVER

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So as you can see, this graph does not cross over the y-axis, so x cannot equal 0.

This is actually why Euler (the inventor of functions, pronounced “Oiler”) also created Domain and Range

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THE DOMAIN AND RANGE OF A FUNCTION

So this is where domain and range came from, to make sure the limitations to a function were met.

So the domain of a function is all of the possible inputs the function can have.

The range of a function is all of the possible outputs the function can have.

An easy way to remember this, is like so:

Inputs are usually x, outputs are usually y.

D comes before R in the alphabet, and X comes before Y in the alphabet.

So domain is all of the x values, and range is all of the y values.

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